arXiv subjects

Alain Connes

Publications and source records attributed to Alain Connes.

At least 73 records · Page 4Linked to original sources

arXiv2013

Modular Curvature for Noncommutative Two-Tori

In this paper we investigate the curvature of conformal deformations by noncommutative Weyl factors of a flat metric on a noncommutative 2-torus, by analyzing in the framework of spectral triples functionals associated to perturbed Dolbeault operators. The analogue of Gaussian curvature turns out to be a sum of two functions in the modular operator corresponding to the non-tracial weight defined by the conformal factor, applied to expressions involving derivatives of the same factor. The first is a generating function for the Bernoulli numbers and is applied to the noncommutative Laplacian of the conformal factor, while the second is a two-variable function and is applied to a quadratic form in the first derivatives of the factor. Further outcomes of the paper include a variational proof of the Gauss-Bonnet theorem for noncommutative 2-tori, the modular analogue of Polyakov's conformal anomaly formula for regularized determinants of Laplacians, a conceptual understanding of the modular curvature as gradient of the Ray-Singer analytic torsion, and the proof using operator positivity that the scale invariant version of the latter assumes its extreme value only at the flat metric.

math.QA↗

arXiv2013

Cyclic structures and the topos of simplicial sets

Given a point p of the topos of simplicial sets and the corresponding flat covariant functor F from the small category Delta to the category of sets, we determine the extensions of F to the cyclic category. We show that to each such cyclic structure on a point p of the topos of simplicial sets corresponds a group G(p), that such groups can be noncommutative and that each G(p) is described as the quotient of a left-ordered group by the subgroup generated by a central element. Moreover for any cyclic set X, the fiber (or geometric realization) of the underlying simplicial set of X at p inherits canonically the structure of a G(p)-space. This gives a far reaching generalization of the well-known circle action on the geometric realization of cyclic sets.

math.AG↗

arXiv2013

Projective geometry in characteristic one and the epicyclic category

We show that the cyclic and epicyclic categories which play a key role in the encoding of cyclic homology and the lambda operations, are obtained from projective geometry in characteristic one over the infinite semifield F of "max-plus integers". Finite dimensional vector spaces are replaced by modules defined by restriction of scalars from the one-dimensional free module, using the Frobenius endomorphisms of F. The associated projective spaces are finite and provide a mathematically consistent interpretation of J. Tits' original idea of a geometry over the absolute point. The self-duality of the cyclic category and the cyclic descent number of permutations both acquire a geometric meaning.

math.AG↗

arXiv2013

Inner Fluctuations in Noncommutative Geometry without the first order condition

We extend inner fluctuations to spectral triples that do not fulfill the first-order condition. This involves the addition of a quadratic term to the usual linear terms. We find a semi-group of inner fluctuations, which only depends on the involutive algebra A and which extends the unitary group of A. This has a key application in noncommutative spectral models beyond the Standard Model, of which we consider here a toy model.

math-ph↗

arXiv2012

Cyclic homology, Serre's local factors and the $λ$-operations

We show that for a smooth, projective variety X defined over a number field K, cyclic homology with coefficients in the ring of infinite adeles of K, provides the right theory to obtain, using the lambda-operations, Serre's archimedean local factors of the complex L-function of X as regularized determinants.

math.AG↗

arXiv2012

Resilience of the Spectral Standard Model

We show that the inconsistency between the spectral Standard Model and the experimental value of the Higgs mass is resolved by the presence of a real scalar field strongly coupled to the Higgs field. This scalar field was already present in the spectral model and we wrongly neglected it in our previous computations. It was shown recently by several authors, independently of the spectral approach, that such a strongly coupled scalar field stabilizes the Standard Model up to unification scale in spite of the low value of the Higgs mass. In this letter we show that the noncommutative neutral singlet modifies substantially the RG analysis, invalidates our previous prediction of Higgs mass in the range 160--180 Gev, and restores the consistency of the noncommutative geometric model with the low Higgs mass.

hep-ph↗

arXiv2011

Spectral Action for Robertson-Walker metrics

We use the Euler-Maclaurin formula and the Feynman-Kac formula to extend our previous method of computation of the spectral action based on the Poisson summation formula. We show how to compute directly the spectral action for the general case of Robertson-Walker metrics. We check the terms of the expansion up to a_6 against the known universal formulas of Gilkey and compute the expansion up to a_{10} using our direct method.

hep-th↗

arXiv2011

On the arithmetic of the BC-system

For each prime p and each embedding of the multiplicative group of an algebraic closure of F_p as complex roots of unity, we construct a p-adic indecomposable representation of the integral BC-system as additive endomorphisms of the big Witt ring of an algebraic closure of F_p. The obtained representations are the p-adic analogues of the complex, extremal KMS states at zero temperature of the BC-system. The role of the Riemann zeta function, as partition function of the BC-system over complex numbers is replaced, in the p-adic case, by the p-adic L-functions and the polylogarithms whose values at roots of unity encode the KMS states. We use Iwasawa theory to extend the KMS theory to a covering of the completion of an algebraic closure of the p-adic field. We show that our previous work on the hyperring structure of the adeles class space, combines with p-adic analysis to refine the space of valuations on the cyclotomic extension of Q as a noncommutative space intimately related to the integral BC-system and whose arithmetic geometry comes close to fulfill the expectations of the "arithmetic site". Finally, we explain how the integral BC-system appears naturally also in de Smit and Lenstra construction of the standard model of an algebraic closure of F_p which singles out the subsystem associated to the Z^-extension of Q.

math.QA↗

arXiv2010

The Witt construction in characteristic one and Quantization

We develop the analogue of the Witt construction in characteristic one. We construct a functor from pairs of a perfect semi-ring of characteristic one and an element strictly larger than one, to real Banach algebras. We find that the entropy function familiar in thermodynamics, ergodic theory and information theory occurs uniquely as the analogue of the Teichmuller polynomials in characteristic one. We then apply the construction to the semi-field of positive real numbers with max as addition, which plays a central role in idempotent analysis and tropical geometry. Our construction gives the inverse process of the ``dequantization" and provides a first hint towards an extension of the field of real numbers relevant both in number theory and quantum physics.

math.QA↗

arXiv2010

Noncommutative Geometric Spaces with Boundary: Spectral Action

We study spectral action for Riemannian manifolds with boundary, and then generalize this to noncommutative spaces which are products of a Riemannian manifold times a finite space. We determine the boundary conditions consistent with the hermiticity of the Dirac operator. We then define spectral triples of noncommutative spaces with boundary. In particular we evaluate the spectral action corresponding to the noncommutative space of the standard model and show that the Einstein-Hilbert action gets modified by the addition of the extrinsic curvature terms with the right sign and coefficient necessary for consistency of the Hamiltonian. We also include effects due to the addition of dilaton field.

hep-th↗

arXiv2010

Space-Time from the spectral point of view

We develop the spectral point of view on geometry based on the formalism of quantum physics. We start from the simple physical question of specifying our position in space and explain how the spectral geometric point of view provides a new paradigm to model space-time whose fine structure can be encoded by a finite geometry. The classification of the irreducible finite geometries of KO-dimension 6 singles out a ``symplectic--unitary" candidate F, which when used as the fine texture of space-time delivers from pure gravity on M x F the Standard Model coupled to gravity and, once extrapolated to unification scale, gives testable predictions.

hep-th↗

arXiv2010

From monoids to hyperstructures: in search of an absolute arithmetic

We show that the trace formula interpretation of the explicit formulas expresses the counting function N(q) of the hypothetical curve C associated to the Riemann zeta function, as an intersection number involving the scaling action on the adele class space. Then, we discuss the algebraic structure of the adele class space both as a monoid and as a hyperring. We construct an extension R^{convex} of the hyperfield S of signs, which is the hyperfield analogue of the semifield R_+^{max} of tropical geometry, admitting a one parameter group of automorphisms fixing S. Finally, we develop function theory over Spec(S) and we show how to recover the field of real numbers from a purely algebraic construction, as the function theory over Spec(S).

math.AG↗

arXiv2010

Noncommutative Geometry as a Framework for Unification of all Fundamental Interactions including Gravity. Part I

We examine the hypothesis that space-time is a product of a continuous four-dimensional manifold times a finite space. A new tensorial notation is developed to present the various constructs of noncommutative geometry. In particular, this notation is used to determine the spectral data of the standard model. The particle spectrum with all of its symmetries is derived, almost uniquely, under the assumption of irreducibility and of dimension 6 modulo 8 for the finite space. The reduction from the natural symmetry group SU(2)xSU(2)xSU(4) to U(1)xSU(2)xSU(3) is a consequence of the hypothesis that the two layers of space-time are finite distance apart but is non-dynamical. The square of the Dirac operator, and all geometrical invariants that appear in the calculation of the heat kernel expansion are evaluated. We re-derive the leading order terms in the spectral action. The geometrical action yields unification of all fundamental interactions including gravity at very high energies. We make the following predictions: (i) The number of fermions per family is 16. (ii) The symmetry group is U(1)xSU(2)xSU(3). (iii) There are quarks and leptons in the correct representations. (iv) There is a doublet Higgs that breaks the electroweak symmetry to U(1). (v) Top quark mass of 170-175 Gev. (v) There is a right-handed neutrino with a see-saw mechanism. Moreover, the zeroth order spectral action obtained with a cut-off function is consistent with experimental data up to few percent. We discuss a number of open issues. We prepare the ground for computing higher order corrections since the predicted mass of the Higgs field is quite sensitive to the higher order corrections. We speculate on the nature of the noncommutative space at Planckian energies and the possible role of the fundamental group for the problem of generations.

hep-th↗

arXiv2010

The hyperring of adèle classes

We show that the theory of hyperrings, due to M. Krasner, supplies a perfect framework to understand the algebraic structure of the adele class space of a global field. After promoting F1 to a hyperfield K, we prove that a hyperring of the form R/G (where R is a ring and G is a subgroup of its multiplicative group) is a hyperring extension of K if and only if G union 0 is a subfield of R. This result applies to the adele class space which thus inherits the structure of a hyperring extension H of K. We begin to investigate the content of an algebraic geometry over K. The category of commutative hyperring extensions of K is inclusive of: commutative algebras over fields with semi-linear homomorphisms, abelian groups with injective homomorphisms and a rather exotic land comprising homogeneous non-Desarguesian planes. Finally, we show that for a global field of positive characteristic, the groupoid of the prime elements of the hyperring H is canonically and equivariantly isomorphic to the groupoid of the loops of the maximal abelian cover of the curve associated to the global field.

math.AG↗

arXiv2009

Characteristic one, entropy and the absolute point

We show that the mathematical meaning of working in characteristic one is directly connected to the fields of idempotent analysis and tropical algebraic geometry and we relate this idea to the notion of the absolute point. After introducing the notion of "perfect" semi-ring of characteristic one, we explain how to adapt the construction of the Witt ring in positive characteristic to the limit case of characteristic one. This construction unveils an interesting connection with entropy and thermodynamics, while shedding a new light on the classical Witt construction itself. We simplify our earlier construction of the geometric realization of an F_1-scheme and extend our earlier computations of the zeta function to cover the case of F_1-schemes with torsion. Then, we show that the study of the additive structures on monoids provides a natural map from monoids to sets which comes close to fulfill the requirements for the hypothetical curve compactifying Spec Z over the absolute point. Finally, we test the computation of the zeta function on elliptic curves over the rational numbers.

math.AG↗

arXiv2009

The Gauss-Bonnet Theorem for the noncommutative two torus

In this paper we show that the value at zero of the zeta function of the Laplacian on the non-commutative two torus, endowed with its canonical conformal structure, is independent of the choice of the volume element (Weyl factor) given by a (non-unimodular) state. We had obtained, in the late eighties, in an unpublished computation, a general formula for this value at zero involving modified logarithms of the modular operator of the state. We give here the detailed computation and prove that the result is independent of the Weyl factor as in the classical case, thus proving the analogue of the Gauss-Bonnet theorem for the noncommutative two torus.

math.QA↗

arXiv2009

Schemes over $\F_1$ and zeta functions

We determine the {\em real} counting function $N(q)$ ($q\in [1,\infty)$) for the hypothetical "curve" $C=\overline {\Sp \Z}$ over $\F_1$, whose corresponding zeta function is the complete Riemann zeta function. Then, we develop a theory of functorial $\F_1$-schemes which reconciles the previous attempts by C. Soulé and A. Deitmar. Our construction fits with the geometry of monoids of K. Kato, is no longer limited to toric varieties and it covers the case of schemes associated to Chevalley groups. Finally we show, using the monoid of adèle classes over an arbitrary global field, how to apply our functorial theory of $\Mo$-schemes to interpret conceptually the spectral realization of zeros of $L$-functions.

math.AG↗

arXiv2009

On the notion of geometry over $\F_1$

We refine the notion of variety over the "field with one element" developed by C. Soulé by introducing a grading in the associated functor to the category of sets, and show that this notion becomes compatible with the geometric viewpoint developed by J. Tits. We then solve an open question of C. Soulé by proving, using results of J. Tits and C. Chevalley, that Chevalley group schemes are examples of varieties over a quadratic extension of the above "field".

math.AG↗